Post-hoc pairwise comparisons are commonly performed after significant effects
have been found when there are three or more levels of a factor. After an ANOVA,
you may know that the means of your response variable differ significantly
across your factor, but you do not know which pairs of the factor levels are
significantly different from each other.  At this point, you can conduct
pairwise comparisons.

We will demonstrate the how to conduct pairwise comparisons in R and the different
options for adjusting the p-values of these comparisons given the number of
tests conducted. We will be using the hsb2 dataset and looking at the
variable write by ses. We will first look at the means and
standard deviations by ses.

hsb2<-read.table("https://stats.idre.ucla.edu/stat/data/hsb2.csv", sep=",", header=T)
attach(hsb2)

ses <- factor(ses) 
levels(ses) <- c("low","medium","high")
female <- factor(female)
levels(female) <- c("male","female")

tapply(write, ses, mean) 

     low   medium     high 
50.61702 51.92632 55.91379 

tapply(write, ses, sd)

     low   medium     high 
9.490391 9.106044 9.442874 

One-Way ANOVA

In R, we can run the ANOVA with the aov command.

a1 <- aov(write ~ ses) 
summary(a1)

             Df Sum Sq Mean Sq F value  Pr(>F)   
ses           2    859   429.4    4.97 0.00784 **
Residuals   197  17020    86.4                   
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

From this output, we can see that ses is significant in the 2-degrees
of freedom test, but we do not know which pairs of ses levels are
significantly different from each other.  However, this will require three
tests (high vs. low, high vs. middle, low vs. middle), so we wish to adjust what
we consider to be statistically significant to account for this multiplicity of
tests.  For an one-way ANOVA (ANOVA with a single factor) We can first see the unadjusted p-values using the
pairwise.t.test command and indicating no adjustment of p-values:

pairwise.t.test(write, ses, p.adj = "none")

        Pairwise comparisons using t tests with pooled SD 

data:  write and ses 

       low    medium
medium 0.4306 -     
high   0.0041 0.0108

P value adjustment method: none

With this same command, we can adjust the p-values according to a variety of methods. Below we show Bonferroni and Holm adjustments to the p-values
and others are detailed in the command help.

pairwise.t.test(write, ses, p.adj = "bonf")

	Pairwise comparisons using t tests with pooled SD 

data:  write and ses 

       low   medium
medium 1.000 -     
high   0.012 0.032 

P value adjustment method: bonferroni

pairwise.t.test(write, ses, p.adj = "holm")

	Pairwise comparisons using t tests with pooled SD 

data:  write and ses 

       low   medium
medium 0.431 -     
high   0.012 0.022 

P value adjustment method: holm 

We can see that the adjustments all lead to increased p-values, but consistently the high-low and high-middle pairs appear to be significantly different at alpha = .05. The pairwise.t.test command does not offer Tukey post-hoc tests, but there are other R commands that allow for Tukey comparisons. Below, we show code for using the TukeyHSD (Tukey Honest Significant Differences).

TukeyHSD(a1)

  Tukey multiple comparisons of means
    95% family-wise confidence level

Fit: aov(formula = write ~ ses)

$ses
                diff        lwr      upr     p adj
medium-low  1.309295 -2.6052575 5.223847 0.7096950
high-low    5.296772  0.9887256 9.604818 0.0114079
high-medium 3.987477  0.3296892 7.645265 0.0289035

We can see that these results are significant with what we saw using other
adjustments for the p-values.

Two (or more) Factor ANOVA

You may be fitting an ANOVA with multiple factors.  Below we look at
write on ses and female.

a2 <- aov(write ~ ses + female)
summary(a2) 

             Df Sum Sq Mean Sq F value   Pr(>F)    
ses           2    859   429.4   5.387  0.00528 ** 
female        1   1398  1398.1  17.541 4.25e-05 ***
Residuals   196  15622    79.7                     
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

We can look at pair-wise comparisons of the ses levels after adjusting
for female. The TukeyHSD command still works well, though now we must
specify which factor is of interest.

TukeyHSD(a2, "ses")

  Tukey multiple comparisons of means
    95% family-wise confidence level

Fit: aov(formula = write ~ ses + female)

$ses
                diff        lwr      upr     p adj
medium-low  1.309295 -2.4507360 5.069325 0.6896199
high-low    5.296772  1.1587797 9.434764 0.0079527
high-medium 3.987477  0.4740753 7.500879 0.0216707